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Properties

Label 2-882-63.59-c1-0-21
Degree $2$
Conductor $882$
Sign $-0.300 - 0.953i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.284 + 1.70i)3-s + (0.499 + 0.866i)4-s + 3.89·5-s + (−1.10 + 1.33i)6-s + 0.999i·8-s + (−2.83 − 0.971i)9-s + (3.36 + 1.94i)10-s + 3.94i·11-s + (−1.62 + 0.608i)12-s + (−2.46 − 1.42i)13-s + (−1.10 + 6.64i)15-s + (−0.5 + 0.866i)16-s + (0.371 − 0.642i)17-s + (−1.97 − 2.26i)18-s + (1.54 − 0.892i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.164 + 0.986i)3-s + (0.249 + 0.433i)4-s + 1.74·5-s + (−0.449 + 0.546i)6-s + 0.353i·8-s + (−0.946 − 0.323i)9-s + (1.06 + 0.615i)10-s + 1.18i·11-s + (−0.468 + 0.175i)12-s + (−0.684 − 0.395i)13-s + (−0.285 + 1.71i)15-s + (−0.125 + 0.216i)16-s + (0.0899 − 0.155i)17-s + (−0.464 − 0.532i)18-s + (0.354 − 0.204i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.300 - 0.953i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56977 + 2.13986i\)
\(L(\frac12)\) \(\approx\) \(1.56977 + 2.13986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.284 - 1.70i)T \)
7 \( 1 \)
good5 \( 1 - 3.89T + 5T^{2} \)
11 \( 1 - 3.94iT - 11T^{2} \)
13 \( 1 + (2.46 + 1.42i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.371 + 0.642i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.54 + 0.892i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.25iT - 23T^{2} \)
29 \( 1 + (2.50 - 1.44i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.04 + 1.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.50 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.24 + 9.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.471 - 0.816i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.09 - 1.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.0105 + 0.0183i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.13 + 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.72 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 + (-4.20 - 2.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.81 - 3.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.02 - 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.63 + 8.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.2 - 9.40i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.21037769922472782550071125242, −9.570441502225982891777526981476, −9.128109106849675071921491426786, −7.67580648313980877222487865246, −6.73530721219092433348725412311, −5.65634585086492426587289588842, −5.29990975078104135210081406680, −4.37206337293430961947673026437, −3.04041063740075612274462613785, −2.00817720661876630546702177348, 1.13495548246155549991736963298, 2.24679645297668556221722972454, 3.00446473001203160625204787959, 4.76832105978630205133655175384, 5.72906181096611984663247308721, 6.18978176184154184950442315795, 6.96099344877620577657971539773, 8.236285693733134740461186248659, 9.127482418275380182123303014572, 10.02784045119783777713657248414

Graph of the $Z$-function along the critical line